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Constitutive behaviors of magnesium and Mg–Zn–Zr alloy during hot deformation
Materials Chemistry and Physics xxx (2014) 1e4
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Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation Hamed Mirzadeh School of Metallurgy and Materials Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Constitutive analysis of pure magnesium and ZK60 alloy during hot compression. Hot deformation activation energy of 135 kJ/mol based on the selfdiffusion of Mg. The theoretical exponent of 5 for the classical hyperbolic sine equation. A constitutive equation with physical and metallurgical backgrounds. A constitutive equation suitable for comparative hot deformation studies.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 April 2014 Received in revised form 3 November 2014 Accepted 15 December 2014 Available online xxx
The flow stress of pure magnesium and ZK60 (Mge6Zn-0.6Zr) magnesium alloy during hightemperature deformation were correlated to the Zener-Hollomon parameter through analyses based on the apparent and physically-based approaches. It was demonstrated that the theoretical exponent of 5 and the lattice self-diffusion activation energy of magnesium (135 kJ/mol) can be set in the hyperbolic sine law to describe the peak flow stress of either the pure Mg or highly alloyed one with zinc and zirconium. As a result, the influence of alloying elements upon the hot flow stress of the ZK60 alloy was characterized by the proposed approach based on the simple material's constants. One of the main prospects of the proposed approach, which is not possible by the conventional approach, is its ability to be utilized in the comparative hot working and alloy development studies. © 2014 Elsevier B.V. All rights reserved.
Keywords: Alloys Hot working Mechanical testing Mechanical properties
1. Introduction The ZK magnesium alloys (MgeZneZr) are among the most commercially important ones. The presence of zirconium in these alloys leads to a fine grain structure, which results in an increase of both strength and formability [1e3]. However, due to their hexagonal closed packed (HCP) crystal structure with a limited number of slip systems, the ductility of polycrystalline Mg and Mg alloys are usually poor at room temperature and hence hot deformation
E-mail address: [email protected].
processing is a suitable shaping method due to the activation of additional slip systems at elevated temperatures [4e6]. Moreover, the structural refinement is another important advantage of hot working [7]. The understanding of the hot working behavior and the constitutive relations describing material flow are two of the prerequisites for the implementation of shaping technology in the industry [8,9]. The modeling of hot flow stress is thus important in metalforming processes because any feasible mathematical simulation needs accurate flow description [10]. A common constitutive equation in hot working is expressed by a hyperbolic sine relation of the form Z ¼ ε_ expðQ =RTÞ ¼ A½sinhðasÞn , where Z (the Zener-
http://dx.doi.org/10.1016/j.matchemphys.2014.12.023 0254-0584/© 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
2
H. Mirzadeh / Materials Chemistry and Physics xxx (2014) 1e4
Hollomon parameter) is the temperature-corrected strain rate, Q is the deformation activation energy, ε_ is the strain rate, T is the absolute temperature, R is the universal gas constant, and finally A (the hyperbolic sine constant), n (the hyperbolic sine power), a (the stress multiplier) are the material's parameters. Conventionally, A, n, a, and Q are considered to be apparent parameters. However, a reliable constitutive equation to characterize the hot deformation behavior will be resulted by consideration of the underlying mechanisms. In the current work, the constitutive behaviors of pure magnesium and one of the most commercially important ZK alloys (i.e. ZK60) with emphasis on the application of physically-based constitutive equations will be analyzed. 2. Experimental materials and procedures The flow stress data of ZK60 alloy with the nominal composition of Mg e 6 wt% Zn e 0.6 wt% Zr [11], hot compressed at deformation temperatures between 200 and 400 C under strain rates of 0.001e1 s1, were taken from the literature [12e15]. More details about these literature data are shown in Table 1. As can be seen in this table, the experimental range of deformation conditions used in the present work is common between most of the considered references. The variations in chemical compositions of the experimental alloys are not significant. Moreover, the initial grain sizes are comparable. It should also be noted that the effect of initial grain size on the flow stress is not significant during hot working and in creep when intragranular dislocation creep is underway [16,17]. The considered flow curves exhibited typical dynamic recrystallization (DRX) behavior with a single peak stress (sP) followed by a gradual fall towards a steady state stress [18]. Note that the description of flow stress by the aforementioned hyperbolic sine equation is incomplete, because no strain for determination of flow stress is specified. Therefore, characteristic stresses that represent the same deformation or softening mechanism for all flow curves, such as steady state, peak, or critical stress for initiation of DRX, should be used in this equation. In general, the peak stress is the most widely accepted one in order to find the values of A, n, a, and Q [19e21]. Therefore, the values of peak stresses were extracted with emphasis on the consistency of stress level among different research works.
hyperbolic sine law (Z ¼ A½sinhðasP Þn ) can be used for a wide range of temperatures and strain rates. In these equations, A0 , A00 , A, n0 , n, b and a (zb/n0 ) are constants. Based on the power and exponential laws, the slopes of the plots of ln_ε against ln sP and ln_ε against sP can be used for obtaining the values of n0 and b, respectively. This is shown in Fig. 1a and the subsequent linear regression of the data resulted in the average value a z 0.010 MPa1. It should be noted that a values of about 0.01 [14], 0.012 [15], and 0.015 MPa1 [22] have also been reported for hot deformation of ZK60 magnesium alloy. 3.2. Consideration of the apparent values of Q and n Taking natural logarithm from the hyperbolic sine equation together with algebraic operations results to Q ¼ R½vln_ε=vlnfsinhðasP ÞgT ½vlnfsinhðasP Þg=vð1=TÞε_ . It follows that the slopes of the plots of ln_ε against lnfsinhðasÞg and lnfsinhðasÞg against 1/T can be used for obtaining the value of Q. The representative plots are shown in Fig. 1b. The linear regression of the data results in the value of Q ¼ 140.3 kJ/mol for the ZK60 alloy. Note that Q values between 115 and 153.3 kJ/mol for the ZK60 alloy [14,15,22] have also been reported. Based on the hyperbolic sine law, the slope and the intercept of
3. Results and discussion 3.1. Determination of the stress multiplier a The Zener-Hollomon parameter (Z) can be related to flow stress 0 in different ways. The power law description of stress (Z ¼ A0 snP ) is preferred for relatively low stresses. Conversely, the exponential 00 law (Z ¼ A expðbsP Þ) is suitable for high stresses. However, the Table 1 Comparison between the experimental set of data used in this study. Reference Chemical composition (wt.%) [12] [13] [14] [15]
Mg e 5.80Zn 0.65Zr Mg e 5.54Zn 0.65Zr Mg e 5.78Zn 0.76Zr Mg e 5.24Zn 0.61Zr
Deformation temperature ( C)
Strain rate Grain size (s1) (mm)
e
150 e 450
e
250 e 450
105 e 101 103 e 10
100
e
200 e 400
103 e 1
70
e
250 e 400
103 e 1
85
120
Fig. 1. Plots used to obtain the values of (a) the stress multiplier a and (b) the deformation activation energy Q for the ZK60 alloy.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
H. Mirzadeh / Materials Chemistry and Physics xxx (2014) 1e4
the plot of ln Z against lnfsinhðasP Þg can be used for obtaining the values of n and A. The corresponding plot is shown in Fig. 2a. The linear regression of the data results in the values of n ¼ 5.43 and A0.2 ¼ 160.55 for the ZK60 alloy. Therefore, a constitutive equation for characterizing the hot deformation behavior of ZK60 alloy based on the apparent values of Q and n can be expressed as follows:
. Z ¼ ε_ exp 140300 RT ¼ 160:555 fsinhð0:01 sP Þg5:43
(1)
3.3. Consideration of the physically-based value of Q The value of hot deformation activation energy (Q ¼ 140.3 kJ/ mol) is close to the value reported for the lattice self-diffusion activation energy of magnesium, which is about 135 kJ/mol [23]. Recently, Mirzadeh et al. [10] have proposed an easy to apply approach that considers theoretical values of n and Q in the constitutive analysis. It has been shown that in hot deformation studies, the self-diffusion activation energy can be used as the deformation activation energy to calculate Z. Therefore, the value of Q ¼ 135 kJ/mol was considered for subsequent constitutive analysis. Again, based on the hyperbolic sine law, the slope and the intercept of the plot of ln Z against lnfsinhðasP Þg can be used for obtaining the values of n and A. The corresponding plot is shown in Fig. 2b. The linear regression of the data results in the values of n ¼ 5.26 and A0.2 ¼ 128.84. Therefore, another constitutive equation for characterizing the hot deformation behavior of ZK60 alloy based on the physically-based value for Q but the apparent value of n can be expressed as follows:
. Z ¼ ε_ exp 135000 RT ¼ 128:845 fsinhð0:01 sP Þg5:26
(2)
3.4. Consideration of the physically-based values of Q and n Eq. (2) shows that the value of the hyperbolic sine power (n) by using the lattice self-diffusion activation energy of magnesium is
Fig. 2. Plots used to obtain the constants of the hyperbolic sine equation for the ZK60 alloy by consideration of (a) apparent values of Q and n, (b) apparent value of n and Q ¼ 135 kJ/mol, and (c) n ¼ 5 and Q ¼ 135 kJ/mol.
3
near 5. Mirzadeh et al. [10] also showed that when the deformation mechanism is controlled by the glide and climb of dislocations, a constant exponent (n) of 5 and self diffusion activation energy can be used to describe the appropriate stress. The obtained values for n and Q in the current study are close to 5 and 135 kJ/mol and this is consistent with the above mentioned metallurgical fact. Therefore, the values of n and Q were considered as 5 and 135 kJ/mol and the result of the corresponding regression analysis is shown in Fig. 2c. The linear regression of the data results in the value of A0.2 ¼ 129.84 for the ZK60 alloy. Therefore, the appropriate and reliable constitutive equation for characterizing the hot deformation behavior of ZK60 alloy, based on the physically-based parameters, can be expressed as follows:
. Z ¼ ε_ exp 135000 RT ¼ 129:845 fsinhð0:01 sP Þg5
(3)
Comparing Eq. (2) with Eq. (1) shows that decreasing the value of Q from 140.3 to 135 kJ/mol results in a decrease in the values of A0.2 and n. Moreover, comparing Eq. (3) with Eq. (2) shows that decreasing the value of n from 5.26 to 5 results in a further decrease in the value of A0.2. The values of the correlation coefficient (R2) for the regression analyses corresponding to Eqs. (1)e(3) are about 0.987, 0.983, and 0.981, respectively. In all of the cases, the values of the correlation coefficient are high and this also justifies the consideration of physically-based material's parameter in constitutive analysis. 3.5. The main prospect of the physically-based approach The proposed approach in the present work considers theoretical values for n and Q in the constitutive analysis. This in turn brings about a possibility to study the constitutive behavior of materials based on the obtained values of A and a. For instance, it is possible to compare the hot flow stress of ZK60 alloy with pure magnesium. The values of the peak flow stress for pure Mg were taken from the literature [24]. The value of a ¼ 0.010 MPa1 was considered and the analyses described in the previous sections were repeated for pure magnesium as shown in Fig. 3. As a result, the apparent value for Q, was determined as 138.4 kJ/mol (Fig. 3a), which is close to the value reported for the lattice self-diffusion activation energy of magnesium (135 kJ/mol). Subsequently, by setting Q ¼ 135 kJ/mol, the value of the hyperbolic sine power was
Fig. 3. Constitutive analyses for hot deformation of pure magnesium.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
4
H. Mirzadeh / Materials Chemistry and Physics xxx (2014) 1e4
determined as n ¼ 5.17 (Fig. 3b). Finally, by setting Q ¼ 135 kJ/mol and n ¼ 5 (Fig. 3c), the appropriate and reliable constitutive equation for characterizing the hot deformation behavior of pure Mg, based on the physically-based parameters, can be expressed as follows:
. Z ¼ ε_ exp 135000 RT ¼ 313:745 fsinhð0:01 sP Þg5
(4)
Comparing Eq. (4) with Eq. (3) reveals that the value of the hyperbolic sine constant A is significantly smaller in the case of the ZK60 alloy. According to the hyperbolic sine law, the flow stress of the material can be expressed as sP ¼ ð1=aÞfsinh1 ðZ=AÞ1=n g or equivalently as sP ¼ ð1=aÞlnfðZ=AÞ1=n þ ½ðZ=AÞ2=n þ 11=2 g. Therefore, it is clear that by decreasing the value of A, the flow stress increases. This is consistent with the fact that addition of the alloying elements to create an alloy generally increases the flow stress of the material by solid solution hardening effects. Therefore, the used approach in the present work can be considered as a versatile tool in future hot working studies, especially in studies dedicating to alloy development. While the consideration of the apparent values of n and Q may result in a better fit to experimental data, but the possibility of elucidating the effects of alloying elements on the hot working behavior will be lost.
4. Conclusions (1) The theoretical value of n ¼ 5 and the lattice self-diffusion activation energy of magnesium (135 kJ/mol) as the hot deformation activation energy (Q) can be used in the classical hyperbolic sine equation to describe the peak stress of either pure Mg or ZK60 magnesium alloy. Therefore, the proposed approach considers theoretical values for n and Q in the constitutive analysis. This in turn brings about a possibility to study the constitutive behavior of materials based on the obtained values of A and a. (2) The equation of Z ¼ ε_ expð135000=RTÞ ¼ Afsinhð0:01 sP Þg5 was proposed to characterize the hot deformation behaviors, where the value of A0.2 is 313.74 and 129.84 for pure Mg and ZK60 alloy, respectively. Since the main difference between these materials is their alloying elements, the difference in constitutive behaviors can be deduced from the values of the constant A. As a result, the proposed approach in the current work can be considered as a versatile tool in comparative hot working studies.
References [1] C.J. Bettles, M.A. Gibson, Current wrought magnesium alloys: strengths and weaknesses, J. Oncol. Manag. 57 (2005) 46e49. [2] Z. Yang, J.P. Li, J.X. Zhang, G.W. Lorimer, J. Robson, Review on research and development of magnesium alloys, Acta Metall. Sin. 21 (2008) 313e328. [3] S.R. Agnew, Wrought magnesium: a 21st century outlook, J. Oncol. Manag. 56 (2004) 20e21. [4] Y.Q. Yang, B.C. Li, Z.M. Zhang, Flow stress of wrought magnesium alloys during hot compression deformation at medium and high temperatures, Mater. Sci. Eng. A 499 (2009) 238e241. [5] S.W. Xu, S. Kamado, N. Matsumoto, T. Honma, Y. Kojima, Recrystallization mechanism of as-cast AZ91 magnesium alloy during hot compressive deformation, Mater. Sci. Eng. A 527 (2009) 52e60. [6] A. El-Morsy, A. Ismail, M. Waly, Microstructural and mechanical properties evolution of magnesium AZ61 alloy processed through a combination of extrusion and thermomechanical processes, Mater. Sci. Eng. A 486 (2008) 528e533. [7] H. Mirzadeh, A. Najafizadeh, Prediction of the critical conditions for initiation of dynamic recrystallization, Mater. Des. 31 (2010) 1174e1179. [8] H. Mirzadeh, A. Najafizadeh, Flow stress prediction at hot working conditions, Mater. Sci. Eng. A 527 (2010) 1160e1164. [9] H. Mirzadeh, M.H. Parsa, D. Ohadi, Hot deformation behavior of austenitic stainless steel for a wide range of initial grain size, Mater. Sci. Eng. A 569 (2013) 54e60. [10] H. Mirzadeh, J.M. Cabrera, A. Najafizadeh, Constitutive relationships for hot deformation of austenite, Acta Mater. 59 (2011) 6441e6448. [11] H.E. Friedrich, B.L. Mordike, Magnesium Technology, Springer, Berlin, 2006. [12] A. Galiyev, R. Kaibyshev, G. Gottstein, Correlation of plastic deformation and dynamic recrystallization in magnesium alloy ZK60, Acta Mater. 49 (2001) 1199e1207. [13] C.Y. Wang, X.J. Wang, H. Chang, K. Wu, M.Y. Zheng, Processing maps for hot working of ZK60 magnesium alloy, Mater. Sci. Eng. A 464 (2007) 52e58. [14] Y.J. Qin, Q.L. Pan, Y.B. He, W.B. Li, X.Y. Liu, X. Fan, Modeling of flow stress for magnesium alloy during hot deformation, Mater. Sci. Eng. A 527 (2010) 2790e2797. [15] H. Yu, H. Yu, G. Min, S.S. Park, B.S. You, Y.M. Kim, Strain-dependent constitutive analysis of hot deformation and hot workability of T4-Treated ZK60 magnesium Alloy, Met. Mater. Int. 19 (2013) 651e665. [16] T.G. Langdon, An analysis of flow mechanisms in high temperature creep and superplasticity, Mater. Trans. 46 (2005) 1951e1956. [17] H. Mirzadeh, Quantification of the strengthening effect of reinforcements during hot deformation of aluminum-based composites, Mater. Des. 65 (2015) 80e82. [18] H. Mirzadeh, J.M. Cabrera, A. Najafizadeh, P.R. Calvillo, EBSD study of a hot deformed austenitic stainless steel, Mater. Sci. Eng. A 538 (2012) 236e245. [19] H. Mirzadeh, J.M. Cabrera, J.M. Prado, A. Najafizadeh, Hot deformation behavior of a medium carbon microalloyed steel, Mater. Sci. Eng. A 528 (2011) 3876e3882. [20] H. Mirzadeh, M.H. Parsa, Hot deformation and dynamic recrystallization of NiTi intermetallic compound, J. Alloys Compd. 614 (2014) 56e59. [21] H. Mirzadeh, A. Najafizadeh, Hot deformation and dynamic recrystallization of 17-4 PH stainless steel, ISIJ Int. 53 (2013) 680e689. [22] S. Spigarelli, M. El Mehtedi, High-temperature deformation and creep in Mg wrought alloys, Scr. Mater. 63 (2010) 704e709. [23] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, second ed., Chapman & Hall, 1992. [24] O. Sitdikov, R. Kaibyshev, Dynamic recrystallization in pure magnesium, Mater. Trans. 42 (2001) 1928e1937.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
Contents lists available at ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation Hamed Mirzadeh School of Metallurgy and Materials Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran
h i g h l i g h t s
g r a p h i c a l a b s t r a c t
Constitutive analysis of pure magnesium and ZK60 alloy during hot compression. Hot deformation activation energy of 135 kJ/mol based on the selfdiffusion of Mg. The theoretical exponent of 5 for the classical hyperbolic sine equation. A constitutive equation with physical and metallurgical backgrounds. A constitutive equation suitable for comparative hot deformation studies.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 April 2014 Received in revised form 3 November 2014 Accepted 15 December 2014 Available online xxx
The flow stress of pure magnesium and ZK60 (Mge6Zn-0.6Zr) magnesium alloy during hightemperature deformation were correlated to the Zener-Hollomon parameter through analyses based on the apparent and physically-based approaches. It was demonstrated that the theoretical exponent of 5 and the lattice self-diffusion activation energy of magnesium (135 kJ/mol) can be set in the hyperbolic sine law to describe the peak flow stress of either the pure Mg or highly alloyed one with zinc and zirconium. As a result, the influence of alloying elements upon the hot flow stress of the ZK60 alloy was characterized by the proposed approach based on the simple material's constants. One of the main prospects of the proposed approach, which is not possible by the conventional approach, is its ability to be utilized in the comparative hot working and alloy development studies. © 2014 Elsevier B.V. All rights reserved.
Keywords: Alloys Hot working Mechanical testing Mechanical properties
1. Introduction The ZK magnesium alloys (MgeZneZr) are among the most commercially important ones. The presence of zirconium in these alloys leads to a fine grain structure, which results in an increase of both strength and formability [1e3]. However, due to their hexagonal closed packed (HCP) crystal structure with a limited number of slip systems, the ductility of polycrystalline Mg and Mg alloys are usually poor at room temperature and hence hot deformation
E-mail address: [email protected].
processing is a suitable shaping method due to the activation of additional slip systems at elevated temperatures [4e6]. Moreover, the structural refinement is another important advantage of hot working [7]. The understanding of the hot working behavior and the constitutive relations describing material flow are two of the prerequisites for the implementation of shaping technology in the industry [8,9]. The modeling of hot flow stress is thus important in metalforming processes because any feasible mathematical simulation needs accurate flow description [10]. A common constitutive equation in hot working is expressed by a hyperbolic sine relation of the form Z ¼ ε_ expðQ =RTÞ ¼ A½sinhðasÞn , where Z (the Zener-
http://dx.doi.org/10.1016/j.matchemphys.2014.12.023 0254-0584/© 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
2
H. Mirzadeh / Materials Chemistry and Physics xxx (2014) 1e4
Hollomon parameter) is the temperature-corrected strain rate, Q is the deformation activation energy, ε_ is the strain rate, T is the absolute temperature, R is the universal gas constant, and finally A (the hyperbolic sine constant), n (the hyperbolic sine power), a (the stress multiplier) are the material's parameters. Conventionally, A, n, a, and Q are considered to be apparent parameters. However, a reliable constitutive equation to characterize the hot deformation behavior will be resulted by consideration of the underlying mechanisms. In the current work, the constitutive behaviors of pure magnesium and one of the most commercially important ZK alloys (i.e. ZK60) with emphasis on the application of physically-based constitutive equations will be analyzed. 2. Experimental materials and procedures The flow stress data of ZK60 alloy with the nominal composition of Mg e 6 wt% Zn e 0.6 wt% Zr [11], hot compressed at deformation temperatures between 200 and 400 C under strain rates of 0.001e1 s1, were taken from the literature [12e15]. More details about these literature data are shown in Table 1. As can be seen in this table, the experimental range of deformation conditions used in the present work is common between most of the considered references. The variations in chemical compositions of the experimental alloys are not significant. Moreover, the initial grain sizes are comparable. It should also be noted that the effect of initial grain size on the flow stress is not significant during hot working and in creep when intragranular dislocation creep is underway [16,17]. The considered flow curves exhibited typical dynamic recrystallization (DRX) behavior with a single peak stress (sP) followed by a gradual fall towards a steady state stress [18]. Note that the description of flow stress by the aforementioned hyperbolic sine equation is incomplete, because no strain for determination of flow stress is specified. Therefore, characteristic stresses that represent the same deformation or softening mechanism for all flow curves, such as steady state, peak, or critical stress for initiation of DRX, should be used in this equation. In general, the peak stress is the most widely accepted one in order to find the values of A, n, a, and Q [19e21]. Therefore, the values of peak stresses were extracted with emphasis on the consistency of stress level among different research works.
hyperbolic sine law (Z ¼ A½sinhðasP Þn ) can be used for a wide range of temperatures and strain rates. In these equations, A0 , A00 , A, n0 , n, b and a (zb/n0 ) are constants. Based on the power and exponential laws, the slopes of the plots of ln_ε against ln sP and ln_ε against sP can be used for obtaining the values of n0 and b, respectively. This is shown in Fig. 1a and the subsequent linear regression of the data resulted in the average value a z 0.010 MPa1. It should be noted that a values of about 0.01 [14], 0.012 [15], and 0.015 MPa1 [22] have also been reported for hot deformation of ZK60 magnesium alloy. 3.2. Consideration of the apparent values of Q and n Taking natural logarithm from the hyperbolic sine equation together with algebraic operations results to Q ¼ R½vln_ε=vlnfsinhðasP ÞgT ½vlnfsinhðasP Þg=vð1=TÞε_ . It follows that the slopes of the plots of ln_ε against lnfsinhðasÞg and lnfsinhðasÞg against 1/T can be used for obtaining the value of Q. The representative plots are shown in Fig. 1b. The linear regression of the data results in the value of Q ¼ 140.3 kJ/mol for the ZK60 alloy. Note that Q values between 115 and 153.3 kJ/mol for the ZK60 alloy [14,15,22] have also been reported. Based on the hyperbolic sine law, the slope and the intercept of
3. Results and discussion 3.1. Determination of the stress multiplier a The Zener-Hollomon parameter (Z) can be related to flow stress 0 in different ways. The power law description of stress (Z ¼ A0 snP ) is preferred for relatively low stresses. Conversely, the exponential 00 law (Z ¼ A expðbsP Þ) is suitable for high stresses. However, the Table 1 Comparison between the experimental set of data used in this study. Reference Chemical composition (wt.%) [12] [13] [14] [15]
Mg e 5.80Zn 0.65Zr Mg e 5.54Zn 0.65Zr Mg e 5.78Zn 0.76Zr Mg e 5.24Zn 0.61Zr
Deformation temperature ( C)
Strain rate Grain size (s1) (mm)
e
150 e 450
e
250 e 450
105 e 101 103 e 10
100
e
200 e 400
103 e 1
70
e
250 e 400
103 e 1
85
120
Fig. 1. Plots used to obtain the values of (a) the stress multiplier a and (b) the deformation activation energy Q for the ZK60 alloy.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
H. Mirzadeh / Materials Chemistry and Physics xxx (2014) 1e4
the plot of ln Z against lnfsinhðasP Þg can be used for obtaining the values of n and A. The corresponding plot is shown in Fig. 2a. The linear regression of the data results in the values of n ¼ 5.43 and A0.2 ¼ 160.55 for the ZK60 alloy. Therefore, a constitutive equation for characterizing the hot deformation behavior of ZK60 alloy based on the apparent values of Q and n can be expressed as follows:
. Z ¼ ε_ exp 140300 RT ¼ 160:555 fsinhð0:01 sP Þg5:43
(1)
3.3. Consideration of the physically-based value of Q The value of hot deformation activation energy (Q ¼ 140.3 kJ/ mol) is close to the value reported for the lattice self-diffusion activation energy of magnesium, which is about 135 kJ/mol [23]. Recently, Mirzadeh et al. [10] have proposed an easy to apply approach that considers theoretical values of n and Q in the constitutive analysis. It has been shown that in hot deformation studies, the self-diffusion activation energy can be used as the deformation activation energy to calculate Z. Therefore, the value of Q ¼ 135 kJ/mol was considered for subsequent constitutive analysis. Again, based on the hyperbolic sine law, the slope and the intercept of the plot of ln Z against lnfsinhðasP Þg can be used for obtaining the values of n and A. The corresponding plot is shown in Fig. 2b. The linear regression of the data results in the values of n ¼ 5.26 and A0.2 ¼ 128.84. Therefore, another constitutive equation for characterizing the hot deformation behavior of ZK60 alloy based on the physically-based value for Q but the apparent value of n can be expressed as follows:
. Z ¼ ε_ exp 135000 RT ¼ 128:845 fsinhð0:01 sP Þg5:26
(2)
3.4. Consideration of the physically-based values of Q and n Eq. (2) shows that the value of the hyperbolic sine power (n) by using the lattice self-diffusion activation energy of magnesium is
Fig. 2. Plots used to obtain the constants of the hyperbolic sine equation for the ZK60 alloy by consideration of (a) apparent values of Q and n, (b) apparent value of n and Q ¼ 135 kJ/mol, and (c) n ¼ 5 and Q ¼ 135 kJ/mol.
3
near 5. Mirzadeh et al. [10] also showed that when the deformation mechanism is controlled by the glide and climb of dislocations, a constant exponent (n) of 5 and self diffusion activation energy can be used to describe the appropriate stress. The obtained values for n and Q in the current study are close to 5 and 135 kJ/mol and this is consistent with the above mentioned metallurgical fact. Therefore, the values of n and Q were considered as 5 and 135 kJ/mol and the result of the corresponding regression analysis is shown in Fig. 2c. The linear regression of the data results in the value of A0.2 ¼ 129.84 for the ZK60 alloy. Therefore, the appropriate and reliable constitutive equation for characterizing the hot deformation behavior of ZK60 alloy, based on the physically-based parameters, can be expressed as follows:
. Z ¼ ε_ exp 135000 RT ¼ 129:845 fsinhð0:01 sP Þg5
(3)
Comparing Eq. (2) with Eq. (1) shows that decreasing the value of Q from 140.3 to 135 kJ/mol results in a decrease in the values of A0.2 and n. Moreover, comparing Eq. (3) with Eq. (2) shows that decreasing the value of n from 5.26 to 5 results in a further decrease in the value of A0.2. The values of the correlation coefficient (R2) for the regression analyses corresponding to Eqs. (1)e(3) are about 0.987, 0.983, and 0.981, respectively. In all of the cases, the values of the correlation coefficient are high and this also justifies the consideration of physically-based material's parameter in constitutive analysis. 3.5. The main prospect of the physically-based approach The proposed approach in the present work considers theoretical values for n and Q in the constitutive analysis. This in turn brings about a possibility to study the constitutive behavior of materials based on the obtained values of A and a. For instance, it is possible to compare the hot flow stress of ZK60 alloy with pure magnesium. The values of the peak flow stress for pure Mg were taken from the literature [24]. The value of a ¼ 0.010 MPa1 was considered and the analyses described in the previous sections were repeated for pure magnesium as shown in Fig. 3. As a result, the apparent value for Q, was determined as 138.4 kJ/mol (Fig. 3a), which is close to the value reported for the lattice self-diffusion activation energy of magnesium (135 kJ/mol). Subsequently, by setting Q ¼ 135 kJ/mol, the value of the hyperbolic sine power was
Fig. 3. Constitutive analyses for hot deformation of pure magnesium.
Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023
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H. Mirzadeh / Materials Chemistry and Physics xxx (2014) 1e4
determined as n ¼ 5.17 (Fig. 3b). Finally, by setting Q ¼ 135 kJ/mol and n ¼ 5 (Fig. 3c), the appropriate and reliable constitutive equation for characterizing the hot deformation behavior of pure Mg, based on the physically-based parameters, can be expressed as follows:
. Z ¼ ε_ exp 135000 RT ¼ 313:745 fsinhð0:01 sP Þg5
(4)
Comparing Eq. (4) with Eq. (3) reveals that the value of the hyperbolic sine constant A is significantly smaller in the case of the ZK60 alloy. According to the hyperbolic sine law, the flow stress of the material can be expressed as sP ¼ ð1=aÞfsinh1 ðZ=AÞ1=n g or equivalently as sP ¼ ð1=aÞlnfðZ=AÞ1=n þ ½ðZ=AÞ2=n þ 11=2 g. Therefore, it is clear that by decreasing the value of A, the flow stress increases. This is consistent with the fact that addition of the alloying elements to create an alloy generally increases the flow stress of the material by solid solution hardening effects. Therefore, the used approach in the present work can be considered as a versatile tool in future hot working studies, especially in studies dedicating to alloy development. While the consideration of the apparent values of n and Q may result in a better fit to experimental data, but the possibility of elucidating the effects of alloying elements on the hot working behavior will be lost.
4. Conclusions (1) The theoretical value of n ¼ 5 and the lattice self-diffusion activation energy of magnesium (135 kJ/mol) as the hot deformation activation energy (Q) can be used in the classical hyperbolic sine equation to describe the peak stress of either pure Mg or ZK60 magnesium alloy. Therefore, the proposed approach considers theoretical values for n and Q in the constitutive analysis. This in turn brings about a possibility to study the constitutive behavior of materials based on the obtained values of A and a. (2) The equation of Z ¼ ε_ expð135000=RTÞ ¼ Afsinhð0:01 sP Þg5 was proposed to characterize the hot deformation behaviors, where the value of A0.2 is 313.74 and 129.84 for pure Mg and ZK60 alloy, respectively. Since the main difference between these materials is their alloying elements, the difference in constitutive behaviors can be deduced from the values of the constant A. As a result, the proposed approach in the current work can be considered as a versatile tool in comparative hot working studies.
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Please cite this article in press as: H. Mirzadeh, Constitutive behaviors of magnesium and MgeZneZr alloy during hot deformation, Materials Chemistry and Physics (2014), http://dx.doi.org/10.1016/j.matchemphys.2014.12.023